Explicit elastic curves
The shape of a bent thin rod (a plane elastic curve) is given explicitly as soon as one knows certain three parameters. The basis of that is a theorem due to the author [cf. \textit{A. Linnér}, Nonlinear Anal., Theory Methods Appl. 21, 575-593 (1993; Zbl 0809.53006), J. Approximation Theory 84, 315-350 (1996; Zbl 0858.41004)], which relates the derivative (with respect to the arc length) of the tangent angle function to an elliptic function, containing three parameters. The parameters are to be found from geometric or elastic properties, in the worst case by nonlinear algebraic equations, but never by differential relations. Special choices of the elliptic functions and of the parameters yield a ``comprehensive, uniform, effective, and efficient'' (according to the author) collection of elastic curves. A number of examples illustrates, case by case, the full range of constraints (open and closed curves, with or without tension, without or with clamping at one or at both of the endpoints). From a differential geometric point of view, this is an interesting collection of plane curves, parametrized by arc length, whose geometric properties are given explicitly by elliptic functions.
scientific article; zbMATH DE number 1211721